A369234 - OEIS

A369234

A longest "diamond" sequence (38 terms). See the Comments section for details.

56, 47, 9, 41, 39, 86, 57, 49, 37, 61, 24, 6, 1, 3, 4, 5, 2, 7, 16, 54, 36, 45, 18, 52, 38, 67, 13, 62, 8, 17, 23, 68, 27, 19, 26, 83, 79, 84

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

All terms are positives and distinct.

No duplicate digit is visible in the set {a(n), a(n+1), a(n)+a(n+1), |a(n)-a(n+1)|}.

If we place the successive sums above the successive commas of the sequence and if we place the successive absolute differences under the said commas, we form three lines of integers that can be seen as a succession of diamonds - hence the name of the sequence.

The "diamond" graph has 59 nodes and 97 edges, where two nodes are linked if they admit a "diamond" (see linked adjacency list). This sequence presents a longest simple path in that graph, namely the lexicographically earliest of the 180 length-38 paths (reversals counted). Longest paths were found using (exhaustive) depth-first search from each start node, remembering paths of longest depth and ties to compute the lexicographically earliest. This was independently confirmed by generating all simple paths in the graph exhaustively, using all_simple_paths in Python's networkx library.

LINKS

Table of n, a(n) for n=1..38.

Éric Angelini and Giorgos Kalogeropoulos, 11 diamonds, personal blog, Feb 2024.

Michael S. Branicky, Adjacency List of "diamond" graph

EXAMPLE

Terms, their sums and their absolute differences begin:

sums 103 56 50 80 125 143

terms 56, 47, 9, 41, 39, 86, 57, ...

diffs 9 38 32 2 47 29

The first diamond is a(1) = 56 and a(2) = 47, with sum 103 and abs diff 9: no digit is duplicated.

The second diamond is a(2) = 47 and a(3) = 9, with sum 56 and abs diff 38: no digit is duplicated. Etc.

CROSSREFS